96 
the angles of crystals. 
In the case of the rhomboid, since the dihedral angles are 
equal, ot,P,ry are equal; and hence also d,e^f are equal. 
Hence 
^ AA'+BB'+Ce-(A' B + AB'+A' C+AC+ B' C + BC') cos. <x 
COS. d = —7 — . 
I (A*4-BHC*-2 (AB + AC+BC) cos.a) (A'»+B'2-|-C»- 2 (A' B'+ A'C-f B'C) cos.«.) j 
And if we put p,q,r, />', f ; r' for A, B, C, A', B', C, we shall 
have the angle. 
If we have to find the angle of two planes resulting from 
the same law, (/; 9'; r') will be a permutation of (/> ; 9 ; r) ; 
and the denominator of — cos. 9 will be 
+ + — 2 (Pq +pr-t-qr) COS. a. 
We shall take examples of the use of these formulae. 
Ej:. 1 . To find the angle made by two planes of carbonate 
of lime resulting from the law* (4, — 5, — 5). {Chaux Car- 
honatee Cuboide of Hauy). 
The primary form of carbonate of lime is a rhomboid in 
which the angle a. is 105® 5', and therefore cos. a = — ,2602. 
Two of the secondary planes will be (4 ; — 5 ; — 5) and 
( — r, ; 4 ; — 5 ), and if 9 be the angle contained by these 
cos. 9 5= 
— 15 — 5 I cos. a 
66 + 30 cos. a 
= 88°. 18. 
or cos. & = 
5 — 17 X .2602 
22 4 10 X .2602 
= .0297 
A variety of other rhomboids may be produced in this and 
other substances by other laws. In all cases, if two of the 
indices of the symbol be equal, as (/>, 9, 9), there will only 
* That this law is what Hauy calls a decrement on the inferior angles of 4 in 
breadth to 5 in height, and is in his notation represented by the symbol e Jl . 
The angles obtained in the text differ slightly from these given by Hauy in con- 
sequence of his having assumed the angle of the primary rhomboid of carbonate of 
lime, = 1040 .28' .40'', for the convenience of using the cosine r: — JL. 
